491 research outputs found
Three-Dimensional Solutions of Supersymmetrical Intertwining Relations and Pairs of Isospectral Hamiltonians
The general solution of SUSY intertwining relations for three-dimensional
Schr\"odinger operators is built using the class of second order supercharges
with nondegenerate constant metric. This solution includes several models with
arbitrary parameters. We are interested only in quantum systems which are not
amenable to separation of variables, i.e. can not be reduced to lower
dimensional problems. All constructed Hamiltonians are partially integrable -
each of them commutes with a symmetry operator of fourth order in momenta. The
same models can be considered also for complex values of parameters leading to
a class of non-Hermitian isospectral Hamiltonians.Comment: 14 page
New Two-Dimensional Quantum Models with Shape Invariance
Two-dimensional quantum models which obey the property of shape invariance
are built in the framework of polynomial two-dimensional SUSY Quantum
Mechanics. They are obtained using the expressions for known one-dimensional
shape invariant potentials. The constructed Hamiltonians are integrable with
symmetry operators of fourth order in momenta, and they are not amenable to the
conventional separation of variables.Comment: 16 p.p., a few new references adde
Matrix Hamiltonians: SUSY approach to hidden symmetries
A new supersymmetric approach to the analysis of dynamical symmetries for
matrix quantum systems is presented. Contrary to standard one dimensional
quantum mechanics where there is no role for an additional symmetry due to
nondegeneracy, matrix hamiltonians allow for non-trivial residual symmetries.
This approach is based on a generalization of the intertwining relations
familiar in SUSY Quantum Mechanics. The corresponding matrix supercharges, of
first or of second order in derivatives, lead to an algebra which incorporates
an additional block diagonal differential matrix operator (referred to as a
"hidden" symmetry operator) found to commute with the superhamiltonian. We
discuss some physical interpretations of such dynamical systems in terms of
spin 1/2 particle in a magnetic field or in terms of coupled channel problem.
Particular attention is paid to the case of transparent matrix potentials.Comment: 20 pages, LaTe
Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction
We study a quantum model with non-isotropic two-dimensional oscillator
potential but with additional quadratic interaction with imaginary
coupling constant. It is shown, that for a specific connection between coupling
constant and oscillator frequences, the model {\it is not} amenable to a
conventional separation of variables. The property of shape invariance allows
to find analytically all eigenfunctions and the spectrum is found to be
equidistant. It is shown that the Hamiltonian is non-diagonalizable, and the
resolution of the identity must include also the corresponding associated
functions. These functions are constructed explicitly, and their properties are
investigated. The problem of separation of variables in two-dimensional
systems is discussed.Comment: 20 pages; minor corrections were made; new Appendix was adde
Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution
We study a class of quantum two-dimensional models with complex potentials of
specific form. They can be considered as the generalization of a recently
studied model with quadratic interaction not amenable to conventional
separation of variables. In the present case, the property of shape invariance
provides the equidistant form of the spectrum and the algorithm to construct
eigenfunctions analytically. It is shown that the Hamiltonian is
non-diagonalizable, and the resolution of identity must include also the
corresponding associated functions. In the specific case of anharmonic
second-plus-fourth order interaction, expressions for the wave functions and
associated functions are constructed explicitly for the lowest levels, and the
recursive algorithm to produce higher level wave functions is given.Comment: 17 p.
New Two-Dimensional Integrable Quantum Models from SUSY Intertwining
Supersymmetrical intertwining relations of second order in the derivatives
are investigated for the case of supercharges with deformed hyperbolic metric
. Several classes of particular solutions of these
relations are found. The corresponding Hamiltonians do not allow the
conventional separation of variables, but they commute with symmetry operators
of fourth order in momenta. For some of these models the specific SUSY
procedure of separation of variables is applied.Comment: 18 page
Relativistic Effects in the Scalar Meson Dynamics
A separable potential formalism is used to describe the and
interactions in the scalar-isoscalar states in the energy range
from the threshold up to 1.4 GeV. Introduction of relativistic
propagators into a system of Lippmann-Schwinger equations leads to a very good
description of the data ( per one degree of freedom). Three
poles are found in this energy region: fo(500) ( MeV,
MeV), fo(975) ( MeV, MeV) and
fo(1400) ( MeV, MeV). The fo(975) state can be
interpreted as a bound state. The fo(500) state may be
associated with the often postulated very broad scalar resonance under the
threshold (sometimes called or meson). The
scattering lengths in the and channels have also been
obtained. The relativistic approach provides qualitatively new results (e.g.
the appearance of the fo(500)) in comparison with previously used
nonrelativistic approach.Comment: 30 pages in LaTeX + 5 figures available on request. Preprint Orsay No
IPNO/TH 93-3
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